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I have already elaborated on uninteresting and interesting labyrinths (see related posts, below). Unintersting labyrinths can be generated by simply attaching additional trivial circuits to the outside or inside of smaller labyrinths. Interesting labyrinths cannot be obtained this way. This particularly implies, that in interesting labyrinths the pathway may not enter the labyrinth on the outermost circuit or reach the center from the innermost circuit. The dual of an interesting labyrinth is another interesting labyrinth too.

This is different if we derive the complementary labyrinth of an interesting labyrinth. The resultung labyrinth may very well be an uninteresting labyrinth. Complementary labyrinths exist only for alternating labyrinths with an odd number of circuits. To obtain the complementary, the pattern of the original labyrinth is vertically mirrored without interrupting the connections between the outside or center of the labyrinth with their corresponding circuits. Labyrinths with an odd number of circuits always have a central circuit. When the pattern is mirrored, this circuit remains in position, whilst the other circuits change their positions symmetrically around it.

Figure 1. Mirroring

In a labyrinth with seven circuits, e.g., the central circuit is the one with number 4. After the mirroring, this remains in its position as number 4. The outermost circuit, number 1, transforms to the innermost circuit and obtains number 7, circuit 2 changes to circuit 6, circuit 3 to circuit 5, and vice versa.

Now, if in an interesting labyrinth the pathway first leads to the innermost circuit or reaches the center from the outermost circuit, then the complementary to this labyrinth will be an uninteresting labyrinth. This, because in the complementary, the path will enter the labyrinth on the outermost or reach the center from the innermost circuit. Thus, there exist pairs of complementary labyrinths, both of which are interesting and others in which one of the labyrinths is interesting and the other uninteresting.

Now I want to find out which are the pairs of interesting complementary labyrinths. The website of Tony Phillips provides best material for such a purpose. On one page, HERE, are included the seed patterns (left figures) and the patterns (right figures) of the interesting alternating labyrinths with up to 7 circuits. I therefore reproduce the page in fig. 2 and in the following add some comments to the items indicated with red letters:

Figure 2. Interesting Labyrinths

  • a) In addition to the circuits, Tony also counts the exterior (= 0) and the center (= one greater than the number of circuits) of a labyrinth. He refers to this as the depth of the labyrinth. A labyrinth with depth 4, thus, has three circuits, one with depth 6 has 5 circuits and so on.
  • b) Below the two figures (seed pattern und pattern), in each case the sequence of circuits is listed. This also contains the zero for the exterior and the number for the center, here indicated with red boxes. The true sequence of circuits is the sequence of numbers between these boxes.
  • c) If the labyrinth is self-dual, this is indicated as „s.d.“ after the sequence of circuits.
  • d) If this is not the case, anyway only one of each dual example is shown in the figures. However, the sequence of circuits of the dual not shown is listed in parentheses below the sequence of circuits of the labyrinth shown.
  • e) The patterns are drawn in such a manner that the course of the pathway leads from top right to bottom left. This is different from how I do it. I draw the pattern from top left to bottom right. As a consequence, the labyrinth that corresponds with the pattern by Tony rotates anti-clockwise, whereas in my case it rotates clockwise.
  • f) Now, lets consider all interesting (including very interesting) labyrinths with 7 circuits. Of these, there are 22 (6 of them very interesting) interesting labyrinths. In fig. 2 the seed patterns and patterns of only 14 labyrinths are depicted. The missing 8, however, are duals, represented by the sequences of circuits in parentheses.

Among the interesting labyrinths with 7 circuits, only 2 exist, in which the pathway does not enter the labyrinth on the innermost circuit nor reach the center from the outermost circuit. And these two form the only pair of interesting labyrinths complementary to each other. We already know this pair from the first post of this series. It is the basic type labyrinth (g) and the labyrinth with the S-shaped course of the pathway (h).

Figure 3. Complementary and Interesting Labyrinths

These are self-dual and thus very interesting labyrinths. In the other 20 interesting labyrinths, the complementary in each case is an uninteresting labyrinth.

Thus, there are 42 different alternating labyrinths with one arm and 7 circuits. Among these, there are 8 pairs of interesting dual labyrinths, 6 self-dual very interesting labyrinths, but only 1 pair of interesting complementary labyrinths. In addition, there is no pair of interesting complementary labyrinths with less than 7 circuits.

Pairs of complementary interesting labyrinths seem to be relatively rare and thus something special.

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It is known that there are 8 alternating labyrinths with 1 arm and 5 circuits (see “Considering Meanders and Labyrinths”, related posts, below). Of these, four are not self-dual. These four all are in a relationship to each other via the duality and complementarity (see “The Complementary versus the Dual Labyrinth”, related posts, below). The other four labyrinths are self-dual.

I had already pointed to the relationship between complementary and self-dual labyrinths (see “The Complementary Labyrinth”, related posts, below). Here I want to elaborate on it further. For this purpose I use the same form of diagram I had already used in my previous post (see “The Complementary versus the Dual Labyrinth”). I also use the same numbers of the labyrinths according to the numbering of Arnol’d’s meanders (see “Considering Meanders and Labyrinths”), that underlie them.

Figure 1. Labyrinths 1 and 6

The first of the Arnol’d’s labyrinths, number 1, is self-dual. In the diagram, the dual is situated in the same row, the complementary in the same column with the original labyrinth. The dual of number 1 is again number 1 (what actually is the meaning of selfdual). The complementary of number 1 is number 6. And – of course – is the dual to the complementary again number 6. So in the case of self-dual labyrinths, we only captured two different labyrinths, whereas it were four in the case of not self-dual labyrinths.

Thus, two more labyrinths are still missing. We need another diagram to capture labyrinths number 3 and number 8 (fig. 2).

Figure 2. Labyrinths 3 and 8

And, indeed, these two are complementary to each other. So in self-dual labyrinths, only two different labyrinths are in a relationship to each other.

Here the question arises: Do there also exist self-complementary labyrinths? Up to now we have not yet found such a labyrinth. So let us remember, what self-dual imples. The patterns of the original and self-dual labyrinths are self-covering. In fig. 3 I show what that means. The two patterns in the same row are dual. If we shift them together, we can easily see, what I mean.

Figure 3. Self-dual patterns are self-covering

Thus, self-complementary would imply that the original and complementary pattern would also be self-covering.

Figure 4. Complementary patterns are not self-covering

Fig. 4 shows, that even though there is a certain similarity between these two patterns, they are not self-covering. In my opinion there are no self-complementary labyrinths. This is because vertical mirroring with uninterrupted connections to the entrance and center modifies the sequence of circuits. This, however, woult have to remain unaltered.

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In the last post I have presented the complementary labyrinth. I did this with the example of the basic type labyrinth. This is a self-dual labyrinth. The complementary is different from the dual labyrinth. This can be better shown using non-self-dual labyrinths. I want tho show this here and for this choose an alternating labyrinth with 1 arm and 5 circuits. As already shown in this blog, there exist 8 such labyrinths (see related post below: Considerung Meanders and Labyrinths). Of these, 4 are self-dual (labyrinths 1, 3, 6, and 8) and 4 are not self-dual (labyrinths 2, 4, 5, and 7).

I thus choose one of the non-self-dual labyrinths, nr. 2, and use the pattern of it. With the pattern, two activities can be performed:

  • Rotate

  • Mirror

Figure 1 shows the result of performing these actions with pattern 2.

Figure 1. Rotating and Mirroring of the Pattern

Rotation leads to the pattern of labyrinth 4
Mirroring leads to pattern 7

So we have already three labyrinths. Now it is possible to go even further. Rotating the dual again brings it back to the original labyrinth. However, the dual can also be mirrored. This results then in the complementary of the dual. And similarly, the complementary can be rotated, which results in the dual to the complementary.

Mirroring of the dual (pattern 4) leads to the complementary pattern of labyrinth 5
Rotation of the complementary (pattern 7) leads to the dual of it – which is also pattern 5.

Figure 2. Relationships

Figure 2 shows the labyrinths corresponding to the patterns. The labyrinths are presented in basic form (i.e shown with their walls delimiting the pathway) in the concentric style. All four non-self-dual alternating labyrinths with 1 arm and 5 circuits are in a relation of either dualtiy or complementarity to each other.

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An interesting labyrinth is reproduced in the book of Kern (fig. 200, p. 119)°. A drawing by Arabian geographer Al Qazvini in his cosmography completed in 1276 is meant to show the ground plan of the residence of the ruler of Byzantium, before the large city of Constantinople was built up.

This non-alternating labyrinth has 10 circuits and a unique course of the pathway. I will show this using the Ariadne’s Thread and the pattern. In my post “From the Ariadne’s Thread to the Pattern – Method 2” (see related posts, below), I have already described how the pattern can be obtained. When deriving the pattern I always start with a labyrinth that rotates clockwise and lies with the entrance from below. The labyrinth by Qazvini rotates in clockwise direction, however it lies with the entrance from above. Therefore I rotate the following images of the labyrinth by a semicircle so that the entrance comes to lie from below. So it is possible to follow the course of the pathway with the Ariadne’s Thread and in parallel see how this is represented in the pattern.

Four steps can be distinguished in the course of the pathway.

Phase 1

The path first leads to the 3rd circuit. The entrance is marked with an arrow pointing inwards. In the pattern, axial sections of the path are represented by vertical, circuits by horizontal lines. The way from the outside in is represented from above to below.

Phase 2

In a second step, the path now winds itself inwards in the shape of a serpentine until it reaches the 10th and innermost circuit. Up to this point the course is alternating.

Phase 3

Next follows the section where the pathway leads from the innermost to the outermost circuit whilst it traverses the axis. In order to derive the pattern, the labyrinth is split along the axis and then uncurled on both sides. As the pathway traverses the axis, the piece of it along the axis has to be split in two halves (see related posts below: “The Pattern in Non-alternating Labyrinths”). This is indicated with the dashed lines. These show one and the same piece of the pathway. In the pattern, as all other axial pieces, this is represented vertically, however with lines showing up on both sides of the rectangular form and a course similarly on both sides from bottom to top.

Phase 4

Finally the pathway continues on the outermost circuit in the same direction it had previously taken on the innermost circuit (anti clockwise), then turns to the second circuit, from where it reaches the center (highlighted with a bullet point).

Related Posts:

°Kern, Hermann. Through the Labyrinth – Designs and Meanings over 5000 Years. Munich: Prestel, 2000.

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Once more: Type in Style

I have now needed three posts to attribute all labyrinth examples of this series to their types. Here I present the last part.

 


Examples in the Reims style

Reims 1

Reims 1

 

 

 

Type Reims


Reims 2

Reims 2

 

 

 

Type Reims


Reims 3

Reims 3

 

 

 

Type Chartres


Reims 4

Reims 4

 

 

 

Type Sneinton (labyrinth drawn faultily)


Reims 5

Reims 5

 

 

 

Type Saffron Walden (labyrinth drawn faultily)


Exemples in the Knidos Style

Knidos 1

Knidos 1

 

 

 

Type Knossos


Knidos 2

Knidos 2

 

 

Core-labyrinth of the type Rockcliffe Marsh, doublespiral-like mander (Erwin’s type 6 meander)


Knidos 3

Knidos 3

 

 

 

 

Cretan type


Knidos 4

Knidos 4

 

 

 

 

Type Otfrid


Other Examples

Andere 1

Other 1

 

 

 

Type Rockcliffe Marsh


Andere 2

Other 2

 

 

 

 

Cretan Type


Andere 3

Other 3

 

 

 

 

Cretan Type


Andere 4

Other 4

 

 

 

 

Type Al Qazwini


Andere 5

Other 5

 

 

 

Type Cakra Vyuh


Andere 6

Other 6

 

 

 

Type Liger


Andere 7

Other 7

 

 

 

Type Ely


Andere 8

Other 8

 

 

 

Type Kieser


Andere 9

Other 9

 

 

 

 

Type Gent


We can see here a similar result as in the two previous posts. The 18 examples belong to 14 different types.
What can be seen here also is, that in some labyrinths the pattern may be difficult to obtain (type Liger, type Ely, type Kieser, type Gent). I do not explain this further here because this is beyond the space of this post.

The types used

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Type in Style (continued)

Here I will now attribute the labyrinth examples from my post Type or Style / 9 to types of labyrinths. These examples were grouped in styles.

The examples in the classical style

Klassisch 1

Classical 1

 

 

 

Type Löwenstein 3


Klassisch 2

Classical 2

 

 

 

Type Löwenstein 5a


Klassisch 3

Classical 3

 

 

 

Cretan Type


Klassisch 4

Classical 4

 

 

 

 

Type Löwenstein 9b


Klassisch 5

Classical 5

 

 

 

Type Hesselager


Klassisch 6

Classical 6

 

 

 

Type Tibble


The examples in the concentric style

Konzentrisch 1

Concentric 1

 

 

 

 

Cretan Type


Konzentrisch 2

Concentric 2

 

 

 

Type Hesselager


Konzentrisch 3

Concentric 3

 

 

 

 

Type Otfrid


Konzentrisch 4

Concentric 4

 

 

 

 

Type Chartres


Konzentrisch 5

Concentric 5

 

 

 

 

Type Gossembrot 51r


Konzentrisch 6

Concentric 6

 

 

 

Type Münster


The examples in the Man-in-the-Maze style

MiM 1

MiM 1

 

 

 

Cretan Type


MiM 2

MiM 2

 

 

 

 

Type Pima


MiM 3, MiM 4: no true labyrinths


The examples in the Chartres style

Chartres 1

Chartres 1

 

 

 

Type Chartres


Chartres 2

Chartres 2

 

 

 

Type Trinity


Chartres 3

Chartres 3

 

 

 

 

Type St. John


Chartres 4

Chartres 4

 

 

 

 

Type Petit Chartres


Chartres 5: no true labyrinth

 

Chartres 6

Chartres 6

 

 

 

Type Grey’s Court


In order not to overload this post I interrupt here and present the remaining examples in my next post.

Among the examples here, there are also figures, that are no unicursal labyrinths in the strict sense. For reasons of space I do not explain this further here but will come back to it later.

The types used:

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Summary

As is the case with the labyrinth itself and the seed pattern, there are also two representations of the rectangular form: this can be represented either with the walls or with the Ariadne’s Thread. In addition, there are two methods to obtain the rectangular form and therefore two versions of it. Ill. 1 summarizes this with the example of my demonstration labyrinth.

L:SP:P Rep

Illustration 1. Overview

This illustration shows on the first line the labyrinth (figures 1), on the second line the seed pattern (figures 2), on the third line the rectangular form obtained with method 1 (figures 3) and on the bottom line the rectangular form obtained with method 2 (figures 4). Each of these are shown in the representation with the walls (left figures a) and with the Ariadne’s Thread (right figures b).

  • When we speak of a „labyrinth“ we usually mean the labyrinth in its representation with the walls. This is shown in fig. 1 a. But also the representation with the Ariadne’s Thread is in widespread use and generally well known (fig. 1 b). This is usually simply referred to as the Ariadne’s Thread.
  • Fig. 2 a shows the seed pattern for the walls, fig. 2 b the seed pattern for the Ariadne’s Thread. As Erwin and I have written so much about this in recent posts, I don’t elaborate more on it here.
  • If we start from the labyrinth (fig. 1 a) or from the Ariadne’s Thread (fig. 1b) and apply method 1, we will as a result obtain the rectangular forms shown in line 3. Thus, there exists a rectangular form for the walls (fig. 3a) as well as for the Ariadne’s Thread (fig. 3b).
  • If we apply method 2 this results in the rectangular forms of line 4. These are the same as the figures on line 3, although rotated by half the arc of a circle.

For what I termed “rectangular form” here, in the literature we can find also the terms „compression diagram“ or „line diagram“ or else. And, most often, we will encounter rectangular forms for the walls obtained with method 1, i.e. figures like fig. 3 a.

RF BM M1

Illustration 2. Figure 3a

I, however, always use the rectangular form for the Ariadne’s Thread. This is the simpler graphical representation. Furthermore, I use the version obtained with method 2, as the result can be read from top left to bottom right, what corresponds better with the way we are used to read. This figure (e.g. fig. 4 b), the rectangular form for the Ariadne’s Thread obtained with method 2, is what I refer to as the pattern.

RF AF M2

Illustration 3. Figure 4b

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